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Mirrors > Home > MPE Home > Th. List > 19.29 | Structured version Visualization version GIF version |
Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1871. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Ref | Expression |
---|---|
19.29 | ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2 472 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
2 | 1 | aleximi 1828 | . 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) |
3 | 2 | imp 409 | 1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 |
This theorem is referenced by: 19.29x 1873 supsrlem 10527 1stccnp 22064 iscmet3 23890 isch3 29012 bnj849 32192 lfuhgr3 32361 axc11n11r 34012 bj-19.42t 34097 stoweidlem35 42314 |
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